The second fundamental form of a parametric surfaceS in R3 was introduced and studied by Gauss. First suppose that the surface is the graph of a twice continuously differentiable function, z = f(x,y), and that the plane z = 0 is tangent to the surface at the origin. Then f and its partial derivatives with respect to x and y vanish at (0,0). Therefore, the Taylor expansion of f at (0,0) starts with quadratic terms:

and the second fundamental form at the origin in the coordinates x, y is the quadratic form

For a smooth point P on S, one can choose the coordinate system so that the coordinate z-plane is tangent to S at P and define the second fundamental form in the same way.

The second fundamental form of a general parametric surface is defined as follows. Let r = r(u,v) be a regular parametrization of a surface in R3, where r is a smooth vector valued function of two variables. It is common to denote the partial derivatives of r with respect to u and v by ru and rv. Regularity of the parametrization means that ru and rv are linearly independent for any (u,v) in the domain of r, and hence span the tangent plane to S at each point. Equivalently, the cross productru × rv is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors n:

The second fundamental form is usually written as

its matrix in the basis {ru, rv} of the tangent plane is

The coefficients L, M, N at a given point in the parametric uv-plane are given by the projections of the second partial derivatives of r at that point onto the normal line to S and can be computed with the aid of the dot product as follows:

The second fundamental form of a general parametric surface S is defined as follows: Let r=r(u1,u2) be a regular parametrization of a surface in R3, where r is a smooth vector valued function of two variables. It is common to denote the partial derivatives of r with respect to uα by rα, α = 1, 2. Regularity of the parametrization means that r1 and r2 are linearly independent for any (u1,u2) in the domain of r, and hence span the tangent plane to S at each point. Equivalently, the cross productr1 × r2 is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors n:

The second fundamental form is usually written as

The equation above uses the Einstein Summation Convention. The coefficients bαβ at a given point in the parametric (u1, u2)-plane are given by the projections of the second partial derivatives of r at that point onto the normal line to S and can be computed in terms of the normal vector "n" as follows:

The sign of the second fundamental form depends on the choice of direction of (which is called a co-orientation of the hypersurface - for surfaces in Euclidean space, this is equivalently given by a choice of orientation of the surface).

For general Riemannian manifolds one has to add the curvature of ambient space; if is a manifold embedded in a Riemannian manifold () then the curvature tensor of with induced metric can be expressed using the second fundamental form and , the curvature tensor of :